Problem: If x x x + 9 3 − x − 9 3 = 3 3 x + 9 ​ − 3 x − 9 ​ = 3 x 2 x 2
Answer Choices:
A. 55 and 65
B. 65 and 75
C. 75 and 85
D. 85 and 95
E. 95 and 105Solution:
x + 9 3 − x − 9 3 = 3 3 x + 9 ​ − 3 x − 9 ​ = 3
x + 9 − 3 ( x + 9 ) 2 / 3 ( x − 9 ) 2 / 3 + 3 ( x + 9 ) 1 / 3 ( x − 9 ) 2 / 3 − x + 9 = 27 x + 9 − 3 ( x + 9 ) 2 / 3 ( x − 9 ) 2 / 3 + 3 ( x + 9 ) 1 / 3 ( x − 9 ) 2 / 3 − x + 9 = 2 7
Simplify to 9 = − 3 ( x + 9 ) 1 / 3 ( x − 9 ) 1 / 3 [ ( x + 9 ) 1 / 3 − ( x − 9 ) 1 / 3 ] = − 3 ( x + 9 ) 2 / 3 ( x − 9 ) 1 / 3 ⋅ 3 9 = − 3 ( x + 9 ) 1 / 3 ( x − 9 ) 1 / 3 [ ( x + 9 ) 1 / 3 − ( x − 9 ) 1 / 3 ] = − 3 ( x + 9 ) 2 / 3 ( x − 9 ) 1 / 3 ⋅ 3
∴ ( x 2 − 81 ) 1 / 3 = − 1 , x 2 = 80 ∴ ( x 2 − 8 1 ) 1 / 3 = − 1 , x 2 = 8 0
or
Let f ( x ) = x + 9 3 − x − 9 3 f ( x ) = 3 x + 9 ​ − 3 x − 9 ​ x x f ( x 1 ) = 3 f ( x 1 ​ ) = 3 f ( 9 ) = 18 3 < 3 f ( 9 ) = 3 1 8 ​ < 3 f ( 8 ) = 17 3 + 1 > 3 f ( 8 ) = 3 1 7 ​ + 1 > 3 8 < x 1 < 9 8 < x 1 ​ < 9 x 1 = 8 7 8 x 1 ​ = 8 8 7 ​ f ( 8 7 8 ) = 17 7 8 3 + 1 2 > 3. ∴ 8 7 8 < x 1 < 9 ∴ ( 8 7 8 ) 2 < x 1 2 < 81 , 78 < x 1 2 < 81 f ( 8 8 7 ​ ) = 3 1 7 8 7 ​ ​ + 2 1 ​ > 3 . ∴ 8 8 7 ​ < x 1 ​ < 9 ∴ ( 8 8 7 ​ ) 2 < x 1 2 ​ < 8 1 , 7 8 < x 1 2 ​ < 8 1